Intuition 1 is incorrect.<p>The probability of the marble being under the box during that period of time was 1 because its state never changed. It was merely your incomplete knowledge that encouraged you to (incorrectly) assign a probability of 0.5. You've become confused over WHAT is the subject of your test.<p>If, however, you were to change the experiment such that there were two boxes, one with a marble and one without, you now have the ability to engage in real probability. The probability that a marble is present under each box is either 1 or 0. However, the probability that you will CHOOSE TO LIFT the box containing the marble is 0.5. This is not a probability of presence or absence of a marble; it's the probability that YOU will choose to look under a particular box.<p>Probability requires the possibility of change, and the probability of change in a marble's presence under a box (such as by teleporting from one box to another) is small indeed.
I understand the question quite differently. The question of objective probabilities is a question of whether the world is deterministic.<p>If you believe the world is governed by physical laws and you further believe you understand those laws (physics), you consequently must believe that with sufficient information about the past you can predict the future (determinism). So if I was to throw a dart at a dartboard, with sufficient information about arm speed, humidity, wind etc you could perfectly predict where on the dartboard it would land. The problem is that the mathematical / computational complexity necessary to achieve that is still beyond our means. Enter Probability, which we use to reduce the computational burden of predicting the future. As probability will always be model dependant (ie subject to someone's view of the world and necessarily wrong some % of the time) - it follows that an objective probability cannot be consistent with a deterministic world.<p>So is the world deterministic?
>To sum up: probabilities are (a) true or false, (b) regardless of what I believe about this probabilities. Estimated probabilities (c) depend on the prior knowledge of the mind in question. Moreover, (d) the fact that the probability is X is a fact about my mind.<p>I'm not sure why you then go on to say that probability is neither objective nor subjective, and that probability has traits of both (which seems somewhat like a contradiction), rather than conclude that there are two different kinds of probability: objective probability and subjective probability ('probabilities' and 'Estimated probabilities' as you even differentiate yourself). (And then there is the other interpretation of the question of whether objective probability even exists at all or wether the world is deterministic.)
On "Intuition 2":<p>> At time t, if it is true that the probability of the marble being under a given box is 0.5, then if I guessed randomly, I would be correct one out of two times on average.<p>> The fact that the probability is 0.5 is completely independent of whether or not any particular mind believes that it is 0.5. But this is the definition of an objective truth. Therefore, probability must be objective.<p>This is begging the question. The two things combined is like saying "At time t, if it is true that probability is objective... therefore probability must be objective."
The question is inherently crippled, because there have to be objective probabilities, otherwise quantum mechanics wouldn't work.<p>I guess the question really is "if the universe operated according to classical principles, would there be objective probabilities?" And that is an interesting question, but it's a hypothetical alternate-universe one.
I parse this question differently. Most commenters seem to be answering 'can ALL priors be objective' when the question is only asking if any exist.<p>For people not sitting this exam, the probability of passing it is zero.
Scanning the exam's questions, I really can't see them as "hard" in the conventional sense.<p>Hard would be a question in math, physics or some other field which has a very difficult to discover objective answer. There's no limit to hard here but these essay questions are different. "Should wealth be inheritable?" just doesn't have an objective answer and one's answer could only be judged based on whatever the exam grader thinks is a good answer (well written, clever, whatever). In fact, considering one can choose three out of twenty seven, the exam doesn't seem much harder than some essay exams I've taken that were supposed to be extremely easy (CBest, for example). Some of the essays would be require to marshal facts in a given field (classics or economics) but we can hope the writers have some expertise in something.<p>I think the only way the questions could be considered hard would be if someone taking it thought they needed to settle one or another of these debatable questions. There you have might the situation where geeks and humanities majors each think the other has "really hard exams"...<p>Edit: And that's "are there objective probabilities" - an English major could take half an hour to answer without worrying about whether they'd really objectively established their answer. We geeks will stop and aim for an "objective" answer. But while one can establish whatever model one wants in the realm of math, one isn't going to pin down the concept of "objective" and "probability" as used by English speaking humans today.
I would say yes, there are objective probabilities. For instance a path integral in quantum mechanics sums all possible paths for a particle along with each path's probability; if the calculations are done correctly, the numbers for a particular calculation will come out the same every time, even if different people perform the calculations. Since the facts and outcomes are observably equal for different people, I would say this fits the definition of objective knowledge. It is what makes quantum mechanics testable and useful.<p>There are plenty of other examples, like the Monty Hall problem, where many different people have confirmed the objective truth of a probability through direct testing.<p>The problem I see with this question is that it is couched as a philosophy question, not math or physics. That's what makes it "hard".